Problem Set 2.pdfProblem Set 2 ECO105 Industrial Organiz.docx

Problem Set 2.pdf
Problem Set 2
ECO105 Industrial Organization and Firm Strategy
Professor Michael Noel
University of California San Diego
------
1. A single-product monopolist sells output to two
geographically separated markets. Arbitrage is not
possible. The inverse demand curve in market 1 is given by p1 =
8 - q1 and in market 2, it is p2 = 4 -
q2. Note: if you get stuck in one part, move on to the next.)
a. The monopolist currently produces the output at a single
plant (called plant "A") with total cost
function C(qA) = 0.5qA
2. Hence qA = q1 + q2 is the total output of the firm (which we
can also
call Q). Find the quantities sold in each market, q1 and q2, and
total output Q. Find the profits of
the firm. (Hint: this is the basic third degree price
discriminating problem. Make sure to
substitute qA out of the problem using the constraint, so that the
problem is all in terms of the
two “free” variables, q1 and q2.)
b. (From an old exam.) The monopolist is considering adding a
second plant, plant "B". The total

cost function for plant "B" is C(qB) = qB + 0.25qB
2. Hence, if the monopolist purchases the new
plant, her total production is qA + qB = Q, where qA is
produced in plant A and qB is produced in
plant B. She then sells Q = q1 + q2 in output, where q1 is sold
in market 1 and q2 is sold in market
2. Find the output sold in each market, q1 and q2, the amount of
good produced at each plant, qA
and qB, and the total output, Q. Find the profits of the firm in
this case. How much is the firm
willing to pay to purchase plant B? (Hint: Do NOT assume that
a single plant is matched solely
with a single market! You only know that the total produced
equals to the total sold, so q1 + q2 =
qA + qB ( = Q). There are now four quantities to find, q1, q2,
qA, qB, but only three of them are
“free variables”, i.e. can be independently chosen. So for
example, if you know q1, q2 and qA, it
necessarily determines what qB must be. So set up your profit
function initially in terms of all
four quantities and but then use the constraint q1 + q2 = qA +
qB to substitute out one of the four.
Which three quantities you keep in the equation is up to you.
Then solve for three first order
conditions.)
c. (Challenging.) Assume a new law prohibits price
discrimination and requires the monopolist to
charge the same price in both markets. Repeat part a. under this
assumption, i.e. only plant A. Has
welfare increased, decreased or remained unchanged relative to
part a.? Warning: check to see if
both markets are being served in equilibrium! (You will need to
“add” up the two demand curves

– add the q’s, not the p’s! A diagram will help. Watch out for
the kinks, it may be that only one
group will be served and it may be that both will be served. At
the end it may help to plot your
numerical answer on your graph and make sure it makes sense.
For example, if you find both
groups are served, you should be on the segment of the demand
curve in your diagram where both
would be served.)
d. Assume instead there is one market with demand given by
p1=2 – q1 and two plants. The cost
function for plant A is C(qA) = qA and the cost function at
plant B is C(qB) = 2qB
2. Solve for the
quantity produced at each plant and the total quantity. (Hint:
this is known as the “multiplant
monopoly” problem. The solution concept is the same as
always: the key thing is to think
carefully about what the profit function must look like. Note
that q1=qA + qB by constraint, so
there are only two “free variables” and you should only be
solving two first order conditions.)
2. A monopolist producing bicycles faces two types of
consumers: light duty users and heavy duty
users, indexed by θL and θ H respectively, where θL < θH.
There are equal numbers of each type of
consumer and each consumer has unit demand for bicycles (ie.
buys one or zero.) A consumer of type

θ has valuation equal to V(θ, q) = 0.5θ(1 – (1-q)2) for a bicycle
of quality q. Marginal cost of
production is constant and equal to c regardless of quality.
a. Find ∂V/∂q, ∂2V/∂q2, ∂V/∂θ, ∂2V/∂q∂θ and verify that the
standard assumptions on these values
hold. Explain the importance of each assumption.
b. Assume the monopolist can observe the type of any given
individual. Set up the fully non-linear
maximization program in which the monopolist offers for sale a
high quality version (q = qH) at a
price RH and a low quality version (q = qL) at a price RL. (For
your practice, do not carry the
general “V” functions throughout the problem (or else you are
just copying from the board) but
instead insert the specific equations for V into your setup of the
problem. You can check your
answers match what we did in class at the end).
c. Solve this maximization program for the optimal qualities
and prices (q L, RL) and (q H, RH).
What is the consumer surplus, also known as “rent”, enjoyed by
the low type? The high type?
Carefully draw the equilibrium in a diagram.
d. Assume instead the monopolist cannot observe the type of
any given individual. Set up the fully
non-linear maximization program in which the monopolist
offers for sale a high quality version
and a low quality version. Plug in the specific equations for V
right from the start. Solve for (q

L, RL) and (q H, RH). What is the consumer surplus, or rent,
enjoyed by the low type? The high
type? Compare the quantities you find to those in part c.)
Carefully draw the equilibrium in a
diagram.
e. Assume instead there are exactly two consumers, person H
and person L, and let VH(q) = 5q and
VL(q) = 4q. (Note that at least one of our usual assumptions on
V is violated…which one(s)?)
Assume the total cost of producing one high quality good and
the low quality good is C(qH) =
qH
2/2 and C(qL) = qL
2/2 respectively. Redo part c and d. with these changes. (Note
that marginal
cost is no longer constant so the equations from the
board…which assumed constant marginal
cost…are no longer identical to what you would get here!)
3. A monopolist faces two types of consumers: low demand
consumers and high demand consumers. A
high demand consumer has valuation equal to VH(q) = 10 + q -
q2 for q units of output and a low
demand consumer has valuation equal to VL(q) = 10 + q - 2q2
for q units of output. There are equal
numbers of each type of consumer. Marginal cost of production
is constant and equal to c. The
monopolist wishes to offer two packages (q L, RL) and (q H,
RH) where qi is the quantity of output

in package i and Ri is the total price for package i.
a. Assuming the monopolist can observe the type of any given
consumer, set up the monopolist’s
fully non-linear maximization program solve this program for
the profit maximizing quantities
and total package prices (q L, RL) and (q H, RH). (Again, insert
your functions for V from the
start.)
b. Assuming the monopolist cannot observe the type of any
given consumer, set up and solve her
maximization program. Compare your answers to part b.)
Problem Set 3.pdf
Problem Set 3
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1. A monopolist produces a durable good. All production takes
place at constant MC= c in the first
period and the good lasts until the end of the second period.
There is a continuum of consumers
uniformly distributed on [0, 1], indexed by θ. A consumer
receives utility u = 2 θ - p if she purchases

the good in the first period (and enjoys it for two periods) and u
= θ - p if she buys in the second
period. The monopolist can charge different prices in each
period.
a. Assume the monopolist can credibly commit to a specific
second period price while still at the
beginning of the first period.
i. Set up the maximization program.
indifferent between buying in the first period
indifferent between buying in the second
and not buying at all.) Warning! – you may be violating an
implicit constraint (chances are, you are.)
Find out what implicit constraint is being broken, then impose
the constraint and solve again.
b. Assuming that the monopolist cannot commit to a second
period price, find θ1, θ 2, q1, p1, q2, and
p2. Which outcome does the monopolist prefer, that of part a. or
part b.?
c. Repeat parts a. and b. but assuming that the utility to a
consumer is u = 3θ- p1 if she purchases in
period 1, u = θ - p2 if she purchases in period 2, 0 if she does
not buy, and MC = 0. (This is
someone who gets extra value from owning something brand

new.)
2. Two firms compete Cournot. The market demand curve is Q =
16/p and each firm faces a cost
function C(qi) = cqi.
a. Find the equilibrium firm quantities, industry quantity, price
and firm profits.
b. Sketch the best response functions for each firm in a single
diagram.
c. Verify that the Lerner Condition (for Cournot competition)
holds.
d. Repeat a. and b. using the market demand curve Q = 1 – p
and a cost function for each firm of
C(qi)=qi/2.
3. There are 9 firms in an industry. The demand curve is given
by Q = 55 - p.
a. Assume each firm has cost function C(qi) = 5qi. (That is,
MCi = 5, for all i = 1..9). Find the total

industry output Q, the price, each firm’s profits and the
Herfindahl index.
b. Assume instead each firm i has cost function C(qi) = iqi, i =
1..9 (That is, MC1 = 1, MC2 = 2,
MC3 = 3,…, MC9 = 9.) Find the total industry output and the
industry price.
c. Is the Herfindahl index higher in part a. or part b.? Is welfare
higher in part a. or part b.? Do not
attempt to do any math here (which is cumbersome) to find
welfare explicitly – use your answers
for Q and P to infer which outcome is more efficient. What does
the asymmetry in costs reveal
about the Herfindahl?
d. Let C(qi) = iqi as in part b. but assume that demand has
fallen to Q = 10 - P. Find total industry
output and industry price. Hint: sum up the first order
conditions. This is tricky -- look carefully
at your answer…are you violating any implicit constraints? If
you are (and you probably are),
impose the constraints and solve again.
4. (From an old exam.) Polar Parkas are sold are two
independent and very distant markets. Market N is
controlled by a profit-maximizing monopolist. In Market S,
there are two Cournot oligopolists:
oligopolist #1 is currently producing twice as much as the
oligopolist #2 in the equilibrium. Also

assume that the marginal cost of oligopolist #1 at its
equilibrium quantity is equal to the marginal cost
of the Market N monopolist at its optimal quantity. Also, at the
equilibrium prices and quantities, the
elasticity of demand in market N is 6 in absolute value, and in
market S the elasticity of demand is 2
in absolute value.
a. Show that the marginal cost of oligopolist #1 is 80% of the
marginal cost of oligopolist #2, at the
equilibrium price and quantities.
b. Show that the price in market N is 80% of the price in market
S in equilibrium. Since market N is
dominated by a monopolist, and market S has competing firms,
explain in a sentence how this
price ordering can be possible?
5. Consider two firms facing a market demand curve p = 2 - Q,
and each having a cost function C(qi) =
qi. (That is, constant MC=1 and no fixed costs.)
a. Assuming the two firms compete Stackelberg where firm 1
plays first, find equilibrium firm
quantities, industry quantity, price and firm profits. Calculate
total welfare (the sum of consumer
surplus and total industry profits). Calculate the market shares
of the two firms.

b. Now assume the cost function for firm 2 has decreased to
C(q2) = 0. The cost function for firm 1
remains C(q1) = q1. Assuming the two firms compete
Stackelberg, find equilibrium firm
quantities, industry quantity, price and firm profits. Calculate
total welfare. Compare your results
from this part with those of part a. If you were a Department of
Justice economist, would you be
concerned about the increased market share of firm 2?
c. Assume instead the market demand curve is P = a – bQ and
costs are zero to each of the two
firms. Assuming Stackelberg competition, solve for the
equilibrium price, firm quantities,
industry quantity, firm profits, consumer surplus, and welfare.
6. Two firms each produce a differentiated good at a constant
marginal cost equal to 1. Firm i faces
demand of qi = 4 + pj - pi, where pi is the price of its own good
and pj is the price of the other firm's
good.
a. If prices are chosen simultaneously, find the best response
functions, equilibrium prices,
quantities and profits.
b. Sketch the best response functions. Label the equilibrium
point E1.

c. If firm 1 is able to credibly commit to choosing its price first
(ie. a Stackelberg leader in this price
game), find the equilibrium prices, quantities and profits.
Compare your results to those of part a.
Who benefits most, the leader or the follower? On the same
diagram, label this equilibrium as
point E2. (Notice that E2 is not on firm 1’s best response
function. Why not?)
d. Repeat parts a. and b. using the demand curve qi = 1 + pj – pi
and assuming costs to each firm are
zero.
7. (From an old exam.) Consider two firms (firm i and firm j)
each selling a differentiated good at zero
marginal cost in the country of Kanada. The demand for the
product sold by firm i is qi = 1 – pi – pj,
with i ≠ j.
a. Are these two goods substitutes or complements?
b. Assuming firms compete by simultaneously choosing its
prices. What are the equilibrium
prices?
c. Now assume that firm 1 and firm 2 enter into a contract in
which they agree fix prices.

Although illegal in the U.S., price fixing contracts are legally
binding in the country of
Kanada. Assuming prices are chosen to maximize industry
profits (and are then split
equally between them), what are the prices that the colluding
firms choose? What’s going
on here?
d. In no more than a few sentences, discuss the welfare
implications of this price-fixing
agreement. (You may calculate welfare in each case if you are
unsure.) Price fixing
agreements are almost always antitcompetitive. But should
Kanada outlaw price fixing
agreements generally? Use your answer to part a. in your
answer here.
Problem Set 4.pdf
Problem Set 4
------
1. Two firms, located at either end of a linear city, compete in
prices. Each firm has constant marginal
cost equal to c. Each consumer i has unit demand, and receives
utility of uij = v - p j - tzij if she buys

from firm j located a distance of zij away at price pj. She
receives utility equal to zero if she does not
buy.
a. If not the whole market is served in equilibrium, find the
equilibrium price and profits for each
firm. Do profits rise or fall with an increase in t?
b. If the whole market is served in equilibrium, find the
equilibrium price and profits for each firm.
Do profits rise or fall when t increases?
c. Give the economic intuition why the sign of dπ/dt is different
in part a. and part
d. Repeat a. and b. using the utility function uij=1-pj-zij and
assuming marginal costs are zero to
each firm. Ignore the questions pertaining to t.
2. Two firms compete in prices along a linear city (as always, of
distance one). One firm is located at the
west end at point 0, the other is located in the center at point ½.
Each firm has constant marginal cost
equal to c. Each consumer i has unit demand, and receives
utility of uij = v - pj - tzij if she buys from
firm j located a distance of zij away at price pj. She receives
utility equal to zero if she does not buy.
Assume that v is large enough that all consumers buy in
equilibrium. Find the equilibrium price and
profits for each firm.

3. Five infinitely lived firms produce homogeneous goods and
operate in a market with demand curve P
= 1 - Q and no marginal or fixed costs of production. Assume
the firms have a cartel agreement in
which they agree to split monopoly profits. Assume each firm
plays a grim trigger strategy: that is, if
any firm cheats, all firms revert back to the Nash equilibrium
strategy of the static game and stay
there forever. For each of the following oligopoly paradigms,
find the range of the discount factor δ
necessary to support the collusive agreement in equilibrium.
a. Firms are Bertrand competitors.
b. Firms are Cournot competitors.
4. Consider two identical, infinitely-lived firms competing in
two markets with identical demand.
However, in market A, each firm can observe in period t the
price its opponent set in period t-1, ie.
cheating can be detected in the next period. In market B, each
firm in period t can only observe past
prices up to (and including) period t-2, ie. a two-period
detection lag. (Thus each firm cannot learn its
opponents price in t-1 until period t+1.) Assume firms compete
in prices. Consider the trigger strategy

in which each firm sets the monopoly price in each market as
long as each has done so in every past
period, but once cheating in either market has been detected by
the opponent, they set price equal to
marginal cost in both markets forever.
a. Warmup: if only market A existed, show that collusion is
sustainable if δ≥1/2.
b. Warmup: if only market B existed, show that collusion is
sustainable if δ≥1/√2.
c. Now assume both markets exist. If a firm were to cheat, what
is the most profitable way to do so?
(Hint: it will cheat as much as possible before being detected.)
Show that collusion is sustainable
if δ = (1 + √17)/8 = 0.64.
5. Assume an incumbent who can choose advertising level K in
the first period. In the second period, the
entrant may enter by paying a fixed cost F. If it enters, the two
firms play Cournot, otherwise the
incumbent is a monopoly. Demand is given by P = K – q1 – q2.
There are no marginal costs. The cost
to the incumbent of K units of advertising is C(K) = K3. (Don’t
forget to include this at the end of the
incumbent’s profit function you write down.)
a. What K does the incumbent choose if entry is blockaded? For
what F is entry blockaded?
b. What K does the incumbent choose if it wishes to deter
entry? What are its profits (as a function
of F) when it deters entry?

c. Does investing in K make the incumbent soft or tough? Does
the incumbent over or underinvest
in K when deterring?
d. What K does the incumbent choose if it accommodates entry?
What are its profits if it
accommodates entry?
e. For which F will the incumbent deter? For which F will the
incumbent accommodate?
f. Does the incumbent over or underinvest when
accommodating? To find out, calculate KOL, the K
that it would choose if the entrant were unable to observe and
react to the incumbents choice of
K. Then your answers to parts d. and e.
g. Let us try to answer e. another way. Calculate the strategic
effect of K on the incumbent’s profits.
Is it positive or negative? Should the incumbent over or
underinvest?
h. Verify that the total derivative of K on the incumbent’s

profits is equal to the direct effect of K on
the incumbent’s profits plus the strategic effect of K on firm 1’s
profits via q2.
i. Let us try to answer e. yet a third way. Can you use the
taxonomy approach (ie. think furry
animals) to decide whether or not the incumbent should over or
underinvest? If not, why not? If
so, which strategy does this game call for and does it call for
over or underinvestment?
6. A new firm (the "entrant", firm 2) is about to enter an
industry currently dominated by a monopolist
(the "incumbent", firm 1.) The entrant does not have to pay a
start up fee so entry cannot be deterred.
When the entrant enters in the second period, the two firms will
produce differentiated products and
compete in prices (chosen simultaneously.) MC = 0 for both
firms. In the first period, before the
entrant gets set up, the incumbent can strategically choose its
advertising level. If the incumbent puts
ads in k newspapers in the first period, its total profits are Π1 =
(3 + 3k - p1 + p2)p1 - 4k
3/3, where the
last term represents the cost of advertising. Profits to the
entrant is given by Π2 = (3 - p2 + p1)p2.
(Note the lack of fixed costs.) The world ends at the end of the
second period.
a. Show that, given the incumbent’s advertising level k,
equilibrium prices in the second period are

p1 = 3 + 2k and p2 = 3 + k. Solve for the profits of each firm in
the second period as a function of
k.
b. What value of k will the incumbent choose in the first period,
knowing the entrant will observe
and react to it in the second period? In other words, what is
kAC?
c. What value of k would the incumbent have chosen in the first
period if the entrant had not been
able to observe and strategically react to the choice of k? That
is, what is kOL?
d. Comparing kAC and kOL from parts b. in c., does the
incumbent overinvest or underinvest in
advertising?
e. Let us answer part d. another way. Calculate the strategic
effect of k on firm 1’s profits. Is it
positive or negative? Hence, does the incumbent overinvest or
underinvest in advertising?
f. Let us answer part d. a third way, using the "taxonomy of

strategies" approach. (Note that direct
effect of k on firm 2’s profits is zero.) Does more advertising
make the incumbent tough or soft?
Are prices strategic substitutes or strategic complements? Given
these answers, what strategy
should the firm use? (Think furry animals.) Does this strategy
call for overinvestment or
underinvestment in advertising?
Problem Set 5.pdf
Problem Set 5
------
1. Consider three equally spaced firms competing in prices
(chosen simultaneously) around a circular
city. There are no fixed costs of production and zero marginal
cost. A consumer receives utility u = v
- p - tz if she purchases from a store a distance of z away, and
you may assume that the parameters are
such that all consumers buy exactly one good in equilibrium
(even after the merger discussed below).
a. Find the prices and profits of each firm, and total industry
profits in equilibrium.
b. Now assume firms 1 and 2 merge. Locations remain the same

and all consumers continue to be
served following the merger.
Find prices, profits to each firm and total industry profits.
Compare to part a. In particular, did
prices rise or fall?
c. If you were a government advisor, would you recommend
preventing this merger if your
objective was low prices?
2. Consider two firms competing in prices (chosen
simultaneously) at opposite ends of a linear city.
There are no fixed costs of production and marginal cost to each
firm is equal to c. A consumer
receives utility u = v - p - tz if she purchases from a store a
distance of z away, and v is so large that
all consumers will buy exactly one good in equilibrium (even
after the merger discussed below).
a. Find prices and the profits to each firm.
b. Show that it would be profitable for the two firms to merge.
c. Show that prices would increase as a result.

d. Imagine the marginal cost of the new firm would be cPOST <
c. Is there a cPOST low enough that
would cause prices to fall after the merger?
3. A common practice for authorities in evaluating mergers is to
observe the stock price responses of
rival firms to the proposed merged firm. What information does
this provide? Is there an incentive for
the rival firms to hide their response or mislead? Discuss why
stock price responses are more
informative to authorities than, for example, lobbying efforts by
the rival firm itself for or against the
merger?

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